Free Electron Model As we mentioned previously, the Pauli exclusion principle plays an important role in the behavior of solids The first quantum model we’ll look at is called the free electron model or the Fermi gas model It’s simple but provides a reasonable description of the properties of the alkalai metals (Li, Na, …) and the nobel metals (Cu, Ag, Au)

Free Electron Model Assume N electrons that are non-interacting (identical) fermions Assume they are confined to a box of V=L3 Hopefully you recall the energy levels of the 3D infinite potential well Note as L grows large the discrete energy spectrum becomes continuous What is the ground state when N is large? Assume T=0 to start with

Free Electron Model Consider a number space to help count the number of states

Free Electron Model Then we can write

Free Electron Model We can then solve for the Fermi energy Many metals have ~1 free electron/ion or n=N/L3 ~ 1029 e/m3 Thus the Fermi energy EF for most metals is ~ 10 eV And the Fermi temperature TF=EF/kB is ~ 105 K Extremely high and much greater than T for room temperature ~ 300 K But this is not the temperature of the electron gas, rather it is a measure of where the Fermi energy is at

Free Electron Model N/L3

Free Electron Model EF and TF

Free Electron Model Comment Once you know the Fermi energy you can also calculate the Fermi momentum and the Fermi velocity The Fermi velocity is the same order of magnitude as the orbital velocity of the outer electrons in an atom and ~10 times the mean thermal velocity of a non-degenerate electron gas It’s still non-relativistic however

Free Electron Model The density of states g(E) is defined as Thus we can calculate n(E)=g(E)FFD

Free Electron Model g(E) and n(E)

Free Electron Model As we learned earlier, once you know n(E) you can calculate many thermodynamic quantities Sanity check

Free Electron Model Total energy I used U here to indicate the total energy

Free Electron Model Recall our discussion earlier in the semester on the molar heat capacity of solids But conduction electrons in a metal should contribute an additional (3/2)R The observed electronic contribution is only ~0.02R

Free Electron Model We can calculate the electronic contribution in the free electron model However up to this point we have been working at T=0 in order to calculate CV so we have to calculate U(T) for T>0

Free Electron Model We could do the integral but it’s more important to notice that in heating a Fermi gas, we populate some states above EF and deplete some states below EF This modification is only important in a narrow energy range kBT around EF kBT

Free Electron Model The fraction of electrons transferred to higher energy is The energy increase of these electrons is So the increase in internal energy is Making the molar heat capacity

Free Electron Model The correct proportionality constant is found doing the integral The point is, that at room temperature, T<<TF and the electronic contribution to the heat capacity is small This is because only a small fraction of electrons around EF can be thermally excited because of the Pauli exclusion principle This was one of the big successes of the free electron model and not predicted classically

Free Electron Model The free electron model for metals can also be used to calculate Electrical conductivity Although to get agreement with experiment, long collision lengths (mean free paths) were required The source of how electrons can move ~ 1 centimeter (for a pure metal at low T) without scattering was unknown Thermal conductivity Although the classical calculation of thermal conductivity (fortuitously) gave good agreement with experiment also